A Lower Bound on the Crossing Number of Uniform Hypergraphs

In this paper, we consider the embedding of a complete $d$-uniform geometric hypergraph with $n$ vertices in general position in $\mathbb{R}^d$, where each hyperedge is represented as a $(d-1)$-simplex, and a pair of hyperedges is defined to cross if they are vertex-disjoint and contains a common point in the relative interior of the simplices corresponding to them. As a corollary of the Van Kampen-Flores Theorem, it can be seen that such a hypergraph contains $\Omega(\frac{2^d}{\sqrt{d}})$ $n\choose 2d$ crossing pairs of hyperedges. Using Gale Transform and Ham Sandwich Theorem, we improve this lower bound to $\Omega(\frac{2^d \log d}{\sqrt{d}})$ $n\choose 2d$.